Modern Control System Theory and Design, 2nd Edition

2.7.  INVERSION BY PARTIAL FRACTION EXPANSION

The time response is the quantity of ultimate interest to the control-system designer. The process of inversion of a function F(s) to find the corresponding time function f(t) is denoted symbolically by

In applications, F(s) is usually a rational function of the form

In practical systems, the order of the polynomial in the denominator is equal to, or greater than, that of the numerator. For the cases where Y > X, partial fraction expansion is directly applicable. When Y X, it is necessary to reduce F(s) to a polynomial in s plus a remainder (ratio of polynomials in s).

The simplest method for obtaining inverse transformations is to use a table of transforms. Unfortunately, many forms of F(s) are not found in the usual table of Laplace-transform pairs. When the form of the solution cannot be readily reduced to a form available in a table, we must use the technique known as partial fraction expansion. This method permits the expansion of the algebraic equation into a series of simpler terms whose transforms are available from a table. It is then possible to obtain the inverse transformation of the original algebraic expression by adding ...